Explicit form of the Hilbert symbol for polynomial formal groups

Abstract:

Let $K$ be a local field, $c$ a unit in $K$, and $F_c (X,Y) = X + Y + cXY$ a polynomial formal group that gives rise to a formal module $F_c(\mathfrak {M})$ on the maximal ideal in the ring of integers of $K$. Assume that $K$ contains the group $\mu _{F_c, n}$ of the roots of isogeny $[p^n]_c(X)$. The natural Hilbert symbol $( \cdot , \cdot )_c \colon K^*\times F_c(\mathfrak {M}) \to \mu _{F_c, n}$ is defined over the module $F(\mathfrak {M})$. An explicit formula for $( \cdot , \cdot )_c$ is constructed.